Nice problem! I considered 1.Ng4+ hxg4 2.d4+ Kf5 but thought it was lost after the queen made it to a6. The problem was keeping the queen out of d3, e2, and f1 which isn't going to happen. White would be put in zugzwang with a simple king move. So 3.Rh1!! (two exclams are fitting!) to set up a stalemate position. Very tough!

And the problem with 1.Nxh5 (h5-h4 IS a threat, opening the position for the queen) is 1...exd3 which opens other avenues of attack. So really, 1.Ng4+ is the only option whether it works or not. The real key is 3.Rh1.

Another point about this problem is that it confounds chess engines! I ran this through Houdini and even after playing over the drawing line, it evaluates it at -5.91. In other words it is saying "Black is up a queen for a rook!" It doesn't go for the draw but instead plays out every possible continuation, avoiding the draw because it can't see the end.